More General Surplus

2016 ◽  
Author(s):  
Alfred Galichon
Keyword(s):  
General Result ◽  
Convex Analysis ◽  
Scalar Product ◽  
Sufficient Conditions ◽  
General Function ◽  
Existence Of A Solution ◽  
Generalize Convex ◽  
Monge Problem ◽  
Natural Way

This chapter considers a case with a more general surplus function. It shows that when the scalar-product surplus is replaced by a more general function, much of the machinery put in place in Chapter 6 goes through. In particular, it is possible to generalize convex analysis in a natural way, and to obtain generalized notions of convex conjugates, of convexity, and of a subdifferential that are perfectly suited to the problem. A general result on the existence of dual minimizers is given, as well as sufficient conditions for the existence of a solution to the Monge problem.

scholarly journals Minimizing spectral risk measures applied to Markov decision processes

2021 ◽  
Author(s):  
Nicole Bäuerle ◽  
Alexander Glauner
Keyword(s):  
Minimization Problem ◽  
Risk Measures ◽  
Risk Measure ◽  
Sufficient Conditions ◽  
Planning Horizon ◽  
Insurance Company ◽  
Existence Of A Solution ◽  
Discounted Cost ◽  
Spectral Risk Measures ◽  
Markov Decision

AbstractWe study the minimization of a spectral risk measure of the total discounted cost generated by a Markov Decision Process (MDP) over a finite or infinite planning horizon. The MDP is assumed to have Borel state and action spaces and the cost function may be unbounded above. The optimization problem is split into two minimization problems using an infimum representation for spectral risk measures. We show that the inner minimization problem can be solved as an ordinary MDP on an extended state space and give sufficient conditions under which an optimal policy exists. Regarding the infinite dimensional outer minimization problem, we prove the existence of a solution and derive an algorithm for its numerical approximation. Our results include the findings in Bäuerle and Ott (Math Methods Oper Res 74(3):361–379, 2011) in the special case that the risk measure is Expected Shortfall. As an application, we present a dynamic extension of the classical static optimal reinsurance problem, where an insurance company minimizes its cost of capital.

scholarly journals The existence of positive solutions for Kirchhoff-type problems via the sub-supersolution method

2018 ◽  
Vol 26 (1) ◽  
pp. 5-41 ◽  
Author(s):  
Baoqiang Yan ◽  
Donal O’Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Positive Solutions ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Existence Of A Solution ◽  
Kirchhoff Type ◽  
Existence Of Positive Solutions ◽  
Global Bifurcation Theory ◽  
Necessary And Sufficient ◽  
Kirchhoff Type Problems

Abstract In this paper we discuss the existence of a solution between wellordered subsolution and supersolution of the Kirchhoff equation. Using the sub-supersolution method together with a Rabinowitz-type global bifurcation theory, we establish the existence of positive solutions for Kirchhoff-type problems when the nonlinearity is singular or sign-changing. Moreover, we obtain some necessary and sufficient conditions for the existence of positive solutions for the problem when N = 1.

scholarly journals Local and global extrema for functions of several variables

1980 ◽  
Vol 29 (3) ◽  
pp. 362-368 ◽  
Author(s):  
Bruce Calvert ◽  
M. K. Vamanamurthy
Keyword(s):  
Critical Point ◽  
Local Minimum ◽  
Global Minimum ◽  
Sufficient Conditions ◽  
Subject Classification ◽  
General Function ◽  
Several Variables ◽  
Open Question

AbstractLet p: R2 → R be a polynomial with a local minimum at its only critical point. This must give a global minimum if the degree of p is < 5, but not necessarily if the degree is ≥ 5. It is an open question what the result is for cubics and quartics in more variables, except cubics in three variables. Other sufficient conditions for a global minimum of a general function are given.1980 Mathematics subject classification (Amer. Math. Soc.): 26 B 99, 26 C 99.

scholarly journals Region of existence of multiple solutions for a class of Robin type four-point BVPs

2021 ◽  
Vol 41 (4) ◽  
pp. 571-600
Author(s):  
Amit K. Verma ◽  
Nazia Urus ◽  
Ravi P. Agarwal
Keyword(s):  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Source Term ◽  
Lipschitz Constant ◽  
Sufficient Conditions ◽  
Existence Of Solution ◽  
Monotone Iterative Technique ◽  
Nonlinear Boundary Value Problems ◽  
Existence Of A Solution ◽  
Monotone Iterative

This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as \[\begin{gathered} -u''(x)=\psi(x,u,u'), \quad x\in (0,1),\\ u'(0)=\lambda_{1}u(\xi), \quad u'(1)=\lambda_{2} u(\eta),\end{gathered}\] where \(I=[0,1]\), \(0\lt\xi\leq\eta\lt 1\) and \(\lambda_1,\lambda_2\gt 0\). The nonlinear source term \(\psi\in C(I\times\mathbb{R}^2,\mathbb{R})\) is one sided Lipschitz in \(u\) with Lipschitz constant \(L_1\) and Lipschitz in \(u'\) such that \(|\psi(x,u,u')-\psi(x,u,v')|\leq L_2(x)|u'-v'|\). We develop monotone iterative technique (MI-technique) in both well ordered and reverse ordered cases. We prove maximum, anti-maximum principle under certain assumptions and use it to show the monotonic behaviour of the sequences of upper-lower solutions. The sufficient conditions are derived for the existence of solution and verified for two examples. The above NLBVPs is linearised using Newton's quasilinearization method which involves a parameter \(k\) equivalent to \(\max_u\frac{\partial \psi}{\partial u}\). We compute the range of \(k\) for which iterative sequences are convergent.

Quadratic Surplus

2016 ◽  
Author(s):  
Alfred Galichon
Keyword(s):  
Convex Analysis ◽  
Scalar Product ◽  
Optimal Transport ◽  
Point Of View ◽  
Factorization Theorem ◽  
Polar Factorization

This chapter considers the case when the attributes are d-dimensional vectors and the surplus is the scalar product; it assumes that the distribution of the workers' attributes is continuous, but it relaxes the assumption that the distribution of the firms' attributes is discrete. This setting allows us to entirely rediscover convex analysis, which is introduced from the point of view of optimal transport. As a consequence, Brenier's polar factorization theorem is given, which provides a vector extension for the scalar notions of quantile and rearrangement.

Approximate controllability of non-instantaneous impulsive semilinear measure driven control system with infinite delay via fundamental solution

2020 ◽  
Author(s):  
Surendra Kumar ◽  
Syed Mohammad Abdal
Keyword(s):  
Fundamental Solution ◽  
Differential Equations ◽  
Control Systems ◽  
Approximate Controllability ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Existence Of A Solution ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
New Class

Abstract This article investigates a new class of non-instantaneous impulsive measure driven control systems with infinite delay. The considered system covers a large class of the hybrid system without any restriction on their Zeno behavior. The concept of measure differential equations is more general as compared to the ordinary impulsive differential equations; consequently, the discussed results are more general than the existing ones. In particular, using the fundamental solution, Krasnoselskii’s fixed-point theorem and the theory of Lebesgue–Stieltjes integral, a new set of sufficient conditions is constructed that ensures the existence of a solution and the approximate controllability of the considered system. Lastly, an example is constructed to demonstrate the effectiveness of obtained results.

scholarly journals Oscillations in a nonautonomous delay logistic difference equation

1992 ◽  
Vol 35 (1) ◽  
pp. 121-131 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Difference Equation ◽  
Equilibrium Point ◽  
Positive Constant ◽  
Sufficient Conditions ◽  
Existence Of A Solution ◽  
Sharp Condition ◽  
All Solutions ◽  
Image Position ◽  
Positive Integers ◽  
Logistic Difference Equation

Consider the nonautonomous delay logistic difference equationwhere (pn)n≧0 is a sequence of nonnegative numbers, (ln)n≧0 is a sequence of positive integers with limn→∞(n−ln) = ∞ and K is a positive constant. Only solutions which are positive for n≧0 are considered. We established a sharp condition under which all solutions of (E0) are oscillatory about the equilibrium point K. Also we obtained sufficient conditions for the existence of a solution of (E0) which is nonoscillatory about K.

On the invertibility of length two elementary operators

2015 ◽  
Vol 30 ◽  
pp. 916-913
Author(s):  
Janko Bracic ◽  
Nadia Boudi
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Elementary Operator ◽  
Bounded Linear Operators ◽  
Existence Of A Solution ◽  
Bounded Linear ◽  
Elementary Operators ◽  
Necessary And Sufficient

Let X be a complex Banach space and L(X) be the algebra of all bounded linear operators on X. For a given elementary operator P of length 2 on L(X), we determine necessary and sufficient conditions for the existence of a solution of the equation YP=0 in the algebra of all elementary operators on L(X). Our approach allows us to characterize some invertible elementary operators of length 2 whose inverses are elementary operators.

scholarly journals Exchange de Risques entre Assureurs et Theorie des Jeux

1977 ◽  
Vol 9 (1-2) ◽  
pp. 155-180 ◽  
Author(s):  
Jean Lemaire
Keyword(s):  
Shapley Value ◽  
Sufficient Conditions ◽  
Pareto Optimal ◽  
Existence Of A Solution ◽  
The Shapley Value ◽  
Théorie Des Jeux ◽  
Person Game ◽  
Reinsurance Market ◽  
Value Concept

SummaryA theorem of Borch characterizing Pareto-optimal treaties in a reinsurance market is extended to the case of non-differentiable utilities. Sufficient conditions for the existence of a solution to the equations are established. The problem is then shown to be identical to the determination of the value of a cooperative non-transferable m-person game. We show how to compute the Shapley value of this game, then we introduce a new value concept. An example illustrates both methods.

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